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Boulder School forCondensed Matter and Materials Physics
Laurette TuckermanPMMH-ESPCI-CNRS
laurette@pmmh.espci.fr
Hydrodynamic Instabilities:
A Zoo of Instabilities
U = Uθ(r) =⇒
−1
rU2
θ = −1
ρdPdr
0 = ν ddr
(
1
rddr
(rUθ)))
Laminar Couette profile: Uθ = Ar +B
r
Rayleigh criterion: laminar Couette profile stable if
d
dr(rUθ)
2 > 0
e.g. if outer cylinder rotates and inner cylinder is stationary
Viscometer (Couette) < Pattern formationstableΩ2 ↑ unstableΩ1 ↑
Taylor-Couette flow
for Ω2 = 0, increasingΩ1
ModulatedLaminar Couette Taylor Vortex Wavy Vortex Wavy Vortex
UC(r) UTV (r, z) UWV (r, θ, z, t) UMWV (r, θ, z, t)
Marangoni-Benard ConvectionTemperature
gradient=⇒
density gradient =⇒ Rayleigh-Benardsurface tension gradient =⇒ Marangoni-Benard
Rayleigh was mistaken about cause of Benard’s observations
Hexagons require breaking of up-down symmetry
Cylinder wake: Benard-von KarmanIdeal flow with downstream recirculation
von Karman vortex street (Re ≥ 46)
Laboratory experiment(Taneda, 1982)
Off Chilean coastpast Juan Fernandez islands
von Karman vortex street: Re = U∞d/ν ≥ 46
spatially: temporally:two-dimensional(x, y) periodic, St = fd/U∞
(homogeneous inz) appears spontaneously
U2D(x, y, t mod T )
Stability analysis of von Karman vortex street
2D limit cycle U2D(x, y, tmod T ) obeys:
∂tU2D = −(U2D · ∇)U2D − ∇P2D +1
Re∆U2D
Perturbation obeys:
∂tu3D = −(U2D(t) · ∇)u3D − (u3D · ∇)U2D(t) − ∇p3D +1
Re∆u3D
Equation homogeneous inz, periodic in t=⇒
u3D(x, y, z, t) ∼ eiβzeλβtfβ(x, y, t mod T )
Fix β, calculate largestµ = eλβT via linearized Navier-Stokes
From Barkley & Henderson, J. Fluid Mech. (1996)
mode A:Rec = 188.5
βc = 1.585 =⇒ λc ≈ 4
mode B:Rec = 259
βc = 7.64 =⇒ λc ≈ 1
Temporally: µ = 1 =⇒
steady bifurcation to limit cycle with same periodicity asU2D
Spatially: circle pitchfork (any phase in z)
mode A atRe = 210 mode B atRe = 250
From M.C. Thompson, Monash University, Australia(http://mec-mail.eng.monash.edu.au/∼mct/mct/docs/cylinder.html)
Faraday instability
Faraday (1831): Vertical vibration of fluid layer =⇒ stripes, squares, hexagons
In 1990s: first fluid-dynamical quasicrystals:
Edwards & Fauve Kudrolli, Pier & GollubJ. Fluid Mech. (1994) Physica D (1998)
Oscillating frame of reference=⇒ “oscillating gravity”
G(t) = g (1 − a cos(ωt)) (1)
G(t) = g (1 − a [cos(mωt) + δ cos(nωt + φ0)]) (2)
Flat surface becomes linearly unstable for sufficiently higha
Consider domain to be horizontally infinite (homogeneous)=⇒solutions exponential/trigonometric inx = (x, y)
Seek bounded solutions=⇒ trigonometric: exp(ik · x)
Height ζ(x, y, t) =∑
k
eik·xζk(t)
Oscillating gravity =⇒ temporal Floquet problem,T = 2π/ω
ζk(t) =∑
j
eλjktf j
k(t)
Height ζ(x, y, t) =∑
k
eik·xζk(t)
Ideal fluids (no viscosity), sinusoidal forcing=⇒ Mathieu eq.
∂2t ζk + ω2
0 [1 − a cos(ωt)] ζk = 0
ω20 combinesg, densities, surface tension, wavenumberk
ζk(t) =2
∑
j=1
eλjktf j
k(t)
Re(λjk) > 1 for somej, k=⇒ ζk ր=⇒ flat surface unstable
=⇒ Faraday waves with wavelength2π/k
Im(λjk) eλT waves period
0 1 harmonic same as forcingπ/ω −1 subharmonic twice forcing period
Floquet functions
XXXXXXXXXXX
XXXX
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X
X
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XXX
X
X
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X
X
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t / T
(ζ-<
ζ>
)/
0 1 2 3
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
λ
XXXXXXXXXXXXXXXXX
X
X
X
X
X
X
XXXX
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XXXXXXXXXXXX
X
X
X
X
X
X
XXXX
XXXXXXXX
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X
X
X
X
X
X
XXXX
XXXt / T
(ζ-<
ζ>
)/
0 1 2 3
-0.002
0
0.002
0.004
0.006
λ
XXXXXXXXXXXXXXXXXXXXXXXXX
X
X
XXXXX
XXXXXXXXXXXXXXXXXXXXXXXX
XXXXXXXXX
XX
X
X
X
XXXXX
XXXXXXXXX
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X
X
X
X
XXXXX
XXXXXXXXXX
t / T
(ζ-<
ζ>
)/
0 1 2 3
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
λ
within tongue 1 /2 within tongue 2 /2 within tongue 3 /2subharmonic harmonic subharmonicµ = −1 µ = +1 µ = −1
From Perinet, Juric & Tuckerman, J. Fluid Mech. (2009)
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